1 On the State-Space Realization of LPV Input-Output ... · systems based on various model...

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On the State-Space Realization of LPV

Input-Output Models: Practical Approaches

R. Toth, Member, IEEE, H. S. Abbas, H. Werner,Member, IEEE

Abstract

A common problem in the context of Linear Parameter-Varying(LPV) systems is how Input-Output

(IO) models can be efficiently realized in terms of State-Space (SS) representations. The problem

originates from the fact that in the LPV literature discrete-time identification and modeling of LPV

systems is often accomplished via IO model structures. However, to utilize these LPV-IO models for

control synthesis, commonly it is required to transform them into an equivalent SS form. In general,

such a transformation is complicated due to the phenomenon of dynamic dependence (dependence

of the resulting representation on time-shifted versions of the scheduling signal). This conversion

problem is revisited and practically applicable approaches are suggested which result in discrete-time

SS representations that have only static dependence (dependence on the instantaneous value of the

scheduling signal). To circumvent complexity, a criterionis also established to decide when an LTI type

of realization approach can be used without introducing significant approximation error. To reduce the

order of the resulting SS realization, a LPV Ho-Kalman type of model reduction approach is introduced,

which, besides its simplicity, is capable of reducing even non-stable plants. The proposed approaches

are illustrated by application oriented examples.

Index Terms

Linear parameter-varying systems; Realization; Model reduction; Input-output representation; State-

space representation; Dynamic dependence.

I. INTRODUCTION

The framework ofLinear Parameter-Varying(LPV) systems provides an efficient alternative

for modeling and control of nonlinear/time-varying systems, proven by a wide range of successful

R. Toth is with the Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The

Netherlands, email: [email protected], H. S. Abbas is withthe Electrical Engineering Department, Faculty of Engineering, Assiut

University, 71515 Assiut, Egypt, email: [email protected] and H. Werner is with the Institute of Control Systems, Hamburg

University of Technology, Eissendorferstr. 40, 21073, Hamburg, Germany, email: [email protected].

January 21, 2011 DRAFT

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applications from aircrafts [1] to environmental modeling[2]. The practical use of the LPV

framework comes from the fact that it offers powerful controller synthesis tools based on the

extension ofLinear Time-Invariant(LTI) approaches, see e.g. [3]–[6].

In the LPV control literature, a discrete-time LPV system iscommonly described in aState-

Space(SS) representation:qx = A(p)x+B(p)u, (1a)

y = C(p)x+D(p)u, (1b)

whereu : Z→ Rnu , y : Z→ R

ny andx : Z→ Rnx are the input, output and state signals of the

system respectively,q is the forward time-shift operator, i.e.qx(k) = x(k + 1), and the system

matricesA,B,C,D are rational functions of the scheduling signalp : Z → P and nonsingular

onP, where the setP ⊆ Rnp is the so calledscheduling space. It is assumed thatp is an external

signal of the system and it is online measurable during operation. In case,p is a function of the

inputs, outputs or states of (1a-b), then the LPV system is referred to as aquasi-LPV system.

Note, that all matrices in (1a-b), defined as

A(p) B(p)

C(p) D(p)

: P→

Rnx×nx R

nx×nu

Rny×nx R

ny×nu

, (2)

are dependent on the instantaneous value ofp, which is calledstatic dependence.

In the LPV literature, numerous approaches have been introduced to identify or to model LPV

systems based on various model structures and representations, see [7] for a recent overview and

comparison of these methods. A large class of the available approaches, like [8]–[11], addresses

the identification problem of LPV systems in terms of so called Input-Output (IO) model

structures with many applied results like [2], [12]–[14]. In this class of LPV-IO identification

methods, the deterministic part of the data-generating system is commonly described in an LPV-

IO filter form:

y = −na∑

i=1

ai(p)q−iy +

nb∑

j=0

bj(p)q−ju, (3)

whereai : P→ Rny×ny andbj : P→ R

ny×nu are rational matrix functions ofp with no singularity

on P and na ≥ nb ≥ 0. However, the main stream of LPV control synthesis approaches is

derived for SS representations, thus to utilize obtained LPV-IO models in the form of (3) for

control synthesis, commonly it is required to transform (3)to an equivalent SS form (1a-b).

Additionally, in terms of LPV modeling based on first-principle nonlinear differential equations


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it is also mathematically attractive to first find a LPV realization of the underlaying behavior in

terms of an IO representation and then find a SS realization ofthe IO form, see [7], [15].

It has been recently observed that representations (1a-b) and (3) are not equivalent in terms

of IO-behavior, i.e. in general, an LPV-IO representation (3) cannot be transformed into (1a-

b) without deforming the dynamical relation ofy and u [16]. This problem, which has been

unknown before, has caused performance loss and significantdifficulties in applications (e.g. see

[12], [13]) as LPV-IO models had been thought to be realizable as LPV-SS models according to

the classical rules of the LTI realization theory. It has been demonstrated in [16] that using such an

intuitive conversion between the two representation structures can lead to40% of output-error

even for slowly varyingp. Furthermore, in the air charge control problem ofSpark Ignition

(SI) gasoline engine, used in this paper for illustration, LPV controllers designed on the SS

realization of an identified high validity LPV-IO model showa significant performance loss if

the SS realization is obtained according to the LTI rules (see Section VI).

Since main-stream LPV controller synthesis approaches arebased on state-space represen-

tations, obtaining SS realization of LPV-IO models has become an essential task to be solved

in practice. According to a recently developed algebraic framework to give a solution for this

transformation problem, see [7], [17], it has been proven that for obtaining equivalence between

SS and IO representations, it is necessary to allow for a dynamic mapping between the scheduling

signals and the system matrices (dynamic dependence), i.e.the system matrices must be allowed

to depend on finite many time-shifted instances ofp(k), like {. . . , p(k− 1), p(k), p(k+1), . . . }.This does increase the complexity of the produced SS model, which may prevent controller

synthesis or hardware implementation of controller designs.

In this paper we propose practical and systematic methods tosolve the problem of transforming

LPV-IO models into LPV-SS forms by avoiding such dynamic dependence on the scheduling

signals. Therefore, we assume that an identified and validated IO model of the system is given for

which realization needs to be addressed, in other words, we deal here only with the realization

problem. Hence, we do not intend to compare performance of identification via LPV-SS or

IO approaches nor posing any of these model structures or methods to be superior above the

other. The developed realization approaches propose a way to close the gap between LPV-

IO modeling/identification and control synthesis. Additionally, a criterion is also established to

decide when the LTI theory inspired realization can be used without introducing a significant


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approximation error. To reduce the order of the resulting SSrealization an LPV Ho-Kalman type

of model reduction approach is introduced, which can even reduce non-stable models. All ideas

are illustrated with simulation studies on practical design examples.

The paper is formulated as follows: In Section II, the transformation problem between LPV-

IO and LPV-SS representations is briefly discussed highlighting the underlaying difficulties and

providing a general solution through an algebraic approachwhich requires the use of dynamic

dependence. Next in Section III, a useful criterion is discussed to decide when the classical LTI

realization theory can be applied without serious consequences and how much can be ’lost’ in

terms of model validity. In Section IV it is discussed in which special cases it is possible to

avoid the introduction of dynamic dependence and to provideexact realization in terms of an

LPV-SS model with static dependence. This is followed in Section V by a brief description of the

LPV Ho-Kalman approach being able to reduce non-minimal SS models resulting from certain

conversion methods. In Section VI, the performance of the proposed approaches is evaluated

through modeling and control design for the intake manifoldof a spark ignition gasoline engine.

Finally, the conclusions of the paper are drawn in Section VII.

II. THE TRANSFORMATION PROBLEM

In the conversion of LPV-IO models to LPV-SS representations it holds true in general that

to preserve IO equivalence of the resulting descriptions, dynamic dependence of the coefficients

on the scheduling signalp (dependence on time-shifted version ofp) must be considered [7],

[17]. To illustrate the problem, investigate the followingsecond-order SS representation:

x1(k + 1)

x2(k + 1)

=

0 a2(p(k))

1 a1(p(k))

x1(k)

x2(k)

+

b2(p(k))

b1(p(k))

u(k),

y(k) = x2(k).

With simple manipulations this system can be written in an equivalent IO form:

y(k) = a1(p(k−1))y(k−1)+a2(p(k−2))y(k−2)+b1(p(k−1))u(k−1)+b2(p(k−2))u(k−2),

which is clearly not in the form of (3) due to the dependence ofthe coefficients onp(k−1) and

p(k−2). We can see from this example that it is necessary to allow fordynamic dependence of the

varying parameters, like state-space matrices or IO coefficients, in order to characterize equivalent

LPV-SS realizations of LPV-IO models. Next, we will investigate how we can reformulate our


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representations to handle such a dynamic dependence in a well-founded sense and how we can

provide algorithms that characterize equivalent representations. For this purpose we will briefly

introduce the algebraic framework of the so called LPVbehavioral approachdeveloped in [17].

A. System representations

In order to describe the functional dependence of a single real-valued coefficient, we employ

functionsr : Rn → R that are considered to be in the fieldR = ∪n∈NRn, whereRn is the set

of essentiallyn-dimensional real-meromorphic functions (being the quotient of analytical real

functions). In this contextessentiallymeans thatr(x1, · · · , xn), wherex = [ x1 . . . xn ] ∈ Rn,

does depend onxn. The functionr specifies how a corresponding coefficient function (like

{A, . . . , D} and{ai, bj} in (1a-b) and (3), respectively) depends onn variables, that are selected

– in a unique ordering – from the set{qipj}i∈Zj=1,··· ,np. More specifically, for a givenP with

dimensionnp and r ∈ Rn, label the variablesx1, . . . , xn of r as ζ0,1, ζ0,1, . . . according to the

following ordering:

r(ζ0,1, . . . , ζ0,np, ζ1,1, . . . , ζ1,np, ζ−1,1, . . . , ζ−1,np, ζ2,1, . . .).

For a given scheduling signalp, associate the variableζi,j with qipj . For this association we

introduce the operator

⋄ : (R,PZ)→ RZ defined by r ⋄ p = r

(

p, qp, q−1p, . . .)

,

whereXZ stands for all maps fromZ to X. Thus the value of a (p-dependent) coefficientr in

an LPV system representation at timek is given by(r ⋄ p)(k).Example 1 (Coefficient function):Let P = R

np with np = 2. Consider the real-meromorphic

coefficient functionr : R3 → R, defined as

r(x1, x2, x3) =1 + x3

1− x2.

Then for a scheduling signalp : Z→ R2:

(r ⋄ p)(k) = r(p1, p2, qp1)(k) =1 + p1(k + 1)

1− p2(k).

On the other hand, ifnp = 3, then (r ⋄ p)(k) = r(p1, p2, p3)(k) = 1+p3(k)1−p2(k)

, showing that the

operator⋄ implicitly depends onnp.

In the sequel the (time-varying) coefficient sequence(r⋄p) will be used to operate on a signalw

(like ai(p) in (3)), giving the varying coefficient sequence of the representations. In this respect


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an important property of the⋄ operation is that multiplication with the shift operatorq is not

commutative, in other wordsq(r ⋄ p)w 6= (r ⋄ p)qw. (4)

To handle this multiplication, forr ∈ R we define the shift operations−→r ,←−r as

−→r = r′ ∈ R s.t. r′ ⋄ p = r ⋄ (qp),←−r = r′′ ∈ R s.t. r′′ ⋄ p = r ⋄ (q−1p),

for all p ∈ (Rnp)Z. With these notions we can writeqr = −→r q and q−1r = ←−r q−1 which

corresponds toq(r ⋄ p)w = (−→r ⋄ p)qw and q−1(r ⋄ p)w = (←−r ⋄ p)q−1w,

in the signal level. This non-commutativity of multiplication with q in the LPV case is the core

problem of realization.

Example 2 (Shift operators):Consider the coefficient functionr given in Example 1 with

np = 2. Then−→r is a functionR5 → R, given by−→r (ζ0,1, ζ0,2, ζ1,1, ζ1,2, ζ−1,1, ζ−1,2, ζ2,1) =1+ζ2,11−ζ1,2

.

For a scheduling trajectoryp : Z→ R2, it holds that(−→r ⋄ p)(k) = (r ⋄ (qp))(k) = 1+p1(k+2)

1−p2(k+1).

Next we can introduce discrete-time LPV-IO and SS representations that have equivalent IO

behavior. LetR[ξ] be the ring of polynomials in the indeterminateξ and with coefficients inR.

Since the indeterminateξ is associated withq, multiplication withξ is non-commutative onR[ξ],i.e. ξr = −→r ξ andrξ = ξ←−r . Then for specified input and output variables(y, u) ∈ (RnY×R

nU)Z

of a given LPV systemS we can introduce theIO representationof S as

(Ry(q) ⋄ p) y = (Ru(q) ⋄ p)u, (5)

whereRy ∈ R[ξ]ny×ny andRu ∈ R[ξ]ny×nu are matrix polynomials with meromorphic coeffi-

cients, e.g.Ry(q) =∑n

i=0 aiqi whereai ∈ R, Ry is full rank anddeg(Ry) ≥ deg(Ru). It is

apparent that (5) is the ‘dynamic-dependent’ counterpart of (3). Furthermore, we can characterize

the solution space of (5) as all maps of(y, u, p) with left-compact support that satisfy (5). We

recognize (5) to be the representation of an LPV systemS if its solution space contains all

trajectories of(y, u, p) that can happen during the operation ofS. Exact characterization of such

behaviors even defining the conditions required forS to be an LPV system are given in [7],

[17].

The natural counterpart of (5) to define SS representation ofan LPV systemS is

(Rw(q) ⋄ p)col(u, y) = (RL(q) ⋄ p)x, (6)


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wherecol(�) denotes the column-vector composition,Rw ∈ R[ξ]nr×(ny+nu) andRL ∈ R[ξ]nr×nx

are matrix polynomials andx is an auxiliary, so calledlatent variable. x satisfies the property

of a state variable if for everyk0 ∈ Z and each solutions(x1, y, u, p) and(x2, y, u, p) of (6) with

x1(k0) = x2(k0) it holds that concatenation of(x1, y, u, p) and(x2, y, u, p) at k0 is a solution of

(6). It can be shown thatx qualifies as a state variable in (6), if and only if (6) can be written

in a form wheredeg(Rw) = 0 anddeg(RL) = 1 [17]. Then the state-space representation can

be formulated as a first-order parameter-varying difference equation system in the state variable

x as

qx = (A ⋄ p)x+ (B ⋄ p)u, (7a)

y = (C ⋄ p)x+ (D ⋄ p)u, (7b)with

A B

C D

∈

Rnx×nx Rnx×nu

Rny×nx Rny×nu

. (8)

It is apparent that (7a-b) are the dynamic-dependent counterparts of (1a-b). It can be shown

that (7a-b) is equivalent with (5) in the sense that for a given LPV-IO representation (5) there

exists an LPV-SS representation (7a-b) with the same IO behavior. The latter means that under

minor restrictions, for all trajectories of(y, u, p) that satisfy (5) there exists ax ∈ (Rnx)Z s.t.

(x, y, u, p) satisfies (7a-b).

B. Equivalent state-space forms

To define equality of LPV-SS and LPV-IO representations, we first have to clarifystate-

transformationsin the LPV case. Consider an LPV-SS representation given by (7a-b). LetT ∈Rnx×nx be invertible (inRnx×nx) and considerx′, given by

x′ = (T ⋄ p)x. (9)

It is immediate that substitution of (9) into (7a) gives

q(T−1 ⋄ p)x′ = (A ⋄ p)(T−1 ⋄ p)x′ + (B ⋄ p)u. (10)

Due to the fact that (10) is a first-order parameter varying difference equation w.r.t.x′, the latent

varaiblex′ trivially qualifies as a new state variable which yields thatan equivalent LPV-SS

representation of (7a-b) reads as

−→T AT−1 −→T BCT−1 D

. (11)


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Similar to the LTI case it can be proven that two LPV state-space representations have the same

IO behavior if and only if their state variables are related via a state-transformation (9). A major

difference w.r.t. LTI state-transformations is that, in the LPV case,T is not only a constant matrix

but can be dependent onp and this dependence can be dynamic, i.e.T ∈ Rnx×nx. Based on the

developed state-transformation and the concept of state-observability and reachability matrices,

the classical canonical forms can also be defined (see [7], [16]).

C. Input-output to state-space

Now the results of [7], [17], which provide the basis for LPV input-output to state-space

transformation, can be formulated as follows:

Theorem 1:Consider a given LPV-IO representation (5) and defineR(ξ) = [ Ry(ξ) −Ru(ξ) ].

Assume that (5) is minimal in the sense thatRy(ξ) andRu(ξ) are coprime. Letw = col(y, u)

and select a full row rankX ∈ R[ξ]·×ny+nu such that for

x = (X(q) ⋄ p)w, (12)

the latent variablex satisfies the property of state, then there exist unique matrix functions

{A,B,C,D} in R·×· and polynomial matrix functionsXu, Xy ∈ R[ξ]·×· with appropriate

dimensions such that

ξX(ξ) = AX(ξ) +BSu +Xu(ξ)R(ξ), (13a)

Sy = CX(ξ) +DSu +Xy(ξ)R(ξ), (13b)

whereSu ∈ Rnu×ny+nu andSy ∈ R

ny×ny+nu are selector matrices1 giving u = Suw andy = Syw.

For a proof see [17]. Note thatX in (12) can be generated fromR by using the so calledcut-

and-shift operations(see [18]). Note also that (13a-b) corresponds to a set of linear equations to

be solved in order to obtain{A,B,C,D} andXu, Xy. With the resulting{A,B,C,D}, (8) is

a minimal (in terms of state-dimension) state-representation of the LPV systemS. In the SISO

case minimality is guaranteed for any choice of full row-rank X satisfying (12), while in the

MIMO case only appropriate selection strategies forX lead to minimal realizations. Furthermore,

specific choices ofX lead to specific canonical forms, see [17] for more details.

For a simple example consider an LPV-IO representation given in the form of

y = pq−1y − pq−2y + pq−2u, (14)

1A matrix with one entry 1 in each row, at most one entry 1 in eachcolumn, and all other entries 0 is a selector matrix.


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with P = R. According to the above theorem, by applying (13a-b) with(X(ξ) ⋄ p) =[

ξ+qp 01 0

]

,

we obtain an LPV-SS realization of (14) in the form of

A B

C D

⋄ p =

0 q2p q2p

1 −qp 0

0 1 0

, (15)

with Xu = [ 1 0 ]⊤ andXy = 0. This LPV-SS form is the so calledcompanion-observability

canonicalform of (14). From this example we can see, that via (13a-b) wecan provide minimal

SS realization of a given LPV-IO representation and in fact via state-transformations we can

characterize all equivalent SS realizations of the system.However, using the results of Theorem

1 we have no control over the introduced scheduling dependence which is most likely to be

dynamic and rational. As for the use of common LPV control synthesis tools most preferably we

need realizations with simple static dependence like linear dependence onp, the basic question

which rises is how we can arrive at a SS realization where alsothe scheduling dependence

is “minimal”. So it is important to explore in which cases we can avoid the use of dynamic

dependence, give direct realization forms and what price wemust pay if we restrict ourselves

to static dependence in terms of an approximative realization. These are the question we intend

to address in the sequel.

III. CRITERION OF DYNAMIC DEPENDENCE

In the literature, the issue of dynamic-dependence of the LPV representation on the scheduling

parameters is often overlooked when an LPV-IO model is transformed into an LPV-SS repre-

sentation, e.g. [13]. Instead, usually LTI realization theory is used to convert (3) to an LPV-SS

form (1a-b) where the matrices have only static dependence.Based on this, (3) is commonly

“realized” in terms of canonical forms, like the reachable (or so called companion reachability)

form given in the SISO case as

A(p) B(p)

C(p) D(p)

=

−a1(p) −a2(p) . . . −ana−1(p) −ana(p) 1

1 0 . . . . . . 0 0

0. . . . . . . . .

......

.... . . . . . 0

......

0 . . . 0 1 0 0

c1(p) . . . . . . . . . cna(p) d(p)

, (16)


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whered = b0 and cj = bj − ajb0 and if nb < na, then bj = 0 for j > nb. In [7], [16], it has

been shown that in the equivalent reachable canonical form of (3), the coefficients have dynamic

dependence onp and they become rational functions of the originalai and bj coefficients of

(3). This implies that (16) is at best an approximation of thetrue canonical realization of the

LPV-IO model, and the introduced approximation error can indeed be arbitrary large [7].

However, dynamic dependence associated with a SS representation of the system commonly

increases the complexity of control synthesis. Thus, it becomes a relevant question when this

approximative realization can be used without serious performance degradation of the designed

controller. A simple answer is to analyze the error between the true and the approximative realiza-

tion. Building upon the realization theory derived in [7], it can be shown that the approximation

error depends on how the Markov parameters{gi}∞i=0 of the plant are approximated. For a given

trajectory ofp denoteh(p, k0)(k) the impulse response of the system at timek for an impulse

applied to the system atk0. Thenh(p, k0)(k) = gk−k0(p, k) for k ≥ k0 andh(p, k0)(k) = 0 for

k < k0. Note that in the LPV case the impulse response of the system depends on the trajectory

of p and the time instance when the impulse is applied. Considering the Markov parameters of

(3) with na = nb = 1 and b0(p) 6= 0, it follows that for a given trajectory ofp

g0(p, k) = b0(pk),

g1(p, k) = −a1(pk)b0(pk−1) + b1(pk), (17)

g2(p, k) = −a1(pk)(

b1(pk−1)− a1(pk−1)b0(pk−2))

,

etc., wherepk = p(k) and it holds true that all solution trajectories of (3) with left compact

support satisfy

y(k) =

∞∑

l=0

gl(p, k)u(k − l). (18)

However, if the LTI realization theory is used, it is assumedthat the Markov parameters are

g0(p, k) = b0(pk),

g1(p, k) = −a1(pk)b0(pk) + b1(pk), (19)

g2(p, k) = −a1(pk)(

b1(pk)− a1(pk)b0(pk))

,

etc. Note the difference of time dependence for each Markov parametergi and gi. As for order

na, the firstna + 1 Markov parameters (with the feedthrough term) completely characterize the


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system dynamics in a functional sense, if

J = supp∈PZ

∥

∥[g0(p, k) . . . gna(p, k)]⊤ − [g0(p, k) . . . gna(p, k)]

⊤∥

∥

∞(20)

is “small,” then the worst-case difference between the IO behavior of the approximative and the

true realization can be considered negligible. In (20),‖ � ‖∞ denotes theℓ∞ norm. However, in

order to quantify what “small” or “acceptable” is from the viewpoint of the user or a particular

application we can introduce a relative thresholdǫ > 0 w.r.t. to theℓ∞ norm of the impulse

response of the original Markov parameter sequence:

J = supp∈PZ

∥

∥[g0(p, k) . . . gna(p, k)]⊤∥

∥

∞. (21)

In this sense, for a givenǫ > 0 and J 6= 0 we can consider the worst-case approximation error

to be acceptable ifJ

J< ǫ. (22)

See Remark 1 for an interpretation of the error in this relative sense. Note thatJ can be computed

in practice by considering the supremum over the values ofg0, . . . , gna for finite sequences

[ p(k) . . . p(k − na) ] = [ p0 . . . pna] ∈ P

na+1. Then by gridding ofPna+1 and assuming

an upper bound on the rate of variation ofp, i.e.‖p(k)−p(k−1)‖ < η, approximate computation

of J becomes available in a lower bound sense. By assuming a set oftrajectoriesP ⊂ PZ of

p that are expected during the operation of the plant, the search space can even be further

decreased.

Example 3:Consider an LPV-IO representation (3) withna = 9, nb = 2, P = [−2π, 0] where

the parameter dependent coefficients are given as follows:

a1(p) = 0.24 + 0.1p, a2(p) = 0.6− 0.1√−p, a3(p) = 0.3 sin(p), a4(p) = 0.17 + 0.1p,

a5(p) = 0.3 cos(p), a6(p) = −0.27, a7(p) = 0.01p, a8(p) = −0.07,

a9(p) = 0.01 cos(p), b0(p) = 1, b1(p) = 1.25− p, b2(p) = −0.2 −√−p.

Note that all coefficients have static dependence onp. A fine grid of P(na+1)=10 is constructed,

where each grid point represents a finite sequence ofp such that‖p(k)−p(k−1)‖ < η1 = 0.01.

Then (20) and (21) are adopted to computeJ and J aiming for ǫ = 1%. The maximum ofJ

over the grid points isJ1 = 0.054, while J = 0.75. SinceJ1 < ǫJ , the LTI realization can be

employed to convert this LPV-IO model withη1 = 0.01 into an adequate LPV-SS form using


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for instance the reachable canonical form. To demonstrate,that satisfying criterion (20) indeed

leads to a SS realization which meets with the expected quality in terms of approximation error,

the best fit rate (BFR) or fit score (see [19])

BFR = 100%.max

(

1− ‖y(k)− y(k)‖2‖y(k)− ym‖2, 0

)

, (23)

where ym is the mean ofy and ‖ � ‖2 is the ℓ2 norm, is used. In other words, theBFR is

applied to validate the obtained approximate LPV-SS model.At each grid point ofP10, (23) is

used to compute the error between the sequence of Markov parametersg0, . . . , gna of the true

LPV-IO model (3) and the sequence of Markov parametersg0(p, k) = D(pk), . . . , gna(p, k) =

C(pk)∏na−1

i=1 A(pk−i)B(pk−na) of the LTI realization provided LPV-SS form with matrix func-

tions{A,B,C,D} having static dependence onp. Note that bothg0, . . . , gna andg0, . . . , gna have

dynamic dependence and{gi} are not the same as in (19). The resulting worst caseBFR over

the grid points isBFR1 = 96.73%. This means that using the LTI realization theory to construct

an LPV-SS form of the original LPV-IO model leads to an acceptable worst-case approximation

error BFR1 if ‖p(k) − p(k − 1)‖ < η1 is satisfied. The example can also be repeated with

η2 = 0.3. According to (20) the maximum achievedJ is J2 = 1.26, so J2 ≫ ǫJ . Therefore

the LTI realization concept in this case is not advised to be used as the resulting error can be

considerable (larger than the specified 1%). This is proven by computing the worst-caseBFR2

over the grid points which is only67.39% in this case.

In order to find a boundaryη for which J < ǫJ , the following problem is solved

η := arg infη≥0

ǫJ − J s.t. ǫJ − J > 0. (24)

By solving (24) in this example, the resultingη is 0.0139 which is the maximum allowed rate of

change ofp, in terms of (21), to guarantee that by applying the LTI realization theory concept

on (3) the resulting SS form will approximate the IO behaviorof (24) adequately (whenǫ is

chosen as1%). The worst case approximation error associated withη is BFR= 96.27% which

can be considered as a good approximation of the original IO behavior of the LPV-IO model.

Remark 1:Note that the considered criterion is applicable both in theSISO and the MIMO

cases. In the later case, the Markov parameters in (20), (21)are matrix functions and hence the

infinity norm of a multidimensional signals is considered. Additionally, the criterion itself can

be formulated in different ways. One can use theℓ2 norm or differentJ according to the specific

application or needs of the user, for instance, the inducedH∞ norm of the error between the


13

two models. With respect to the latter case it can be shown that for an asymptotically stable

LPV system, the inducedH∞ norm satisfies

sup‖u‖2<∞u 6=0, p∈PZ

‖∑∞i=0 gi(p)q

−iu‖2‖u‖2

≤ supp∈PZ

k∈Z

‖h(p, k)‖2. (25)

This shows that the criterionJ ≤ ǫJ in terms of (20) with anℓ2 norm and with an adequately

large number of Markov parameters can be interpreted as bounding theH∞ norm of the error in

a relative sense. However, the error bound is delivered w.r.t. the assumption of static dependence

of the Markov parameters and not directly the error between the LTI realization provided SS

model and (3).

IV. DEDICATED LPV-SS REALIZATIONS TO ENSURE STATIC DEPENDENCE

By considering the realization theory briefly discussed in Section II and fully developed in

[7], it can be shown that in special cases there exist ways of converting LPV-IO models to

LPV-SS realizations without introducing dynamic dependence. However, each realization form

is either based on specific assumptions or provides non-minimal SS realizations, which can be

later converted into minimal realizations using appropriate tools, see Section V.

A. Shifted form

For the sake of simplicity we consider only the SISO case. However, the realization forms

that will be introduced can be extended to the MIMO case in a straightforward manner similar

to the LTI case. Assume that the LPV-IO model is given in the form

y +na∑

i=1

ai(q−ip)q−iy =

nb∑

j=0

bj(q−jp)q−ju, (26)

whereai, bj : P→ R. Note thatai andbj have a special form of dynamic dependence which can

be introduced into the parametrization of most of the available LPV-IO identification approaches.

Now introducex1 asy = x1 + b0(p)u, (27a)

qx1 =

nb−1∑

j=0

bj+1(q−jp)q−ju−

na−1∑

i=0

ai+1(q−ip)q−iy. (27b)

Continue the so called natural state construction (see [7])as

qx1 = x2 − a1(p)y + b1(p)u, (28a)

qx2 =

nb−2∑

j=0

bj+2(q−jp)q−ju−

na−2∑

i=0

ai+2(q−ip)q−iy. (28b)


14

until qxna = −ana(p)y + bna(p)u. Thus we obtain a set of first-order difference equations:

q

x1......

xna−1

xna

=

0 1 0 . . . 0...

. . . . . . . . ....

.... . . . . . . . .

...

0 . . . . . . 0 1

0 . . . . . . . . . 0

x1......

xna−1

xna

+

−a1(p) b1(p)...

......

...

−ana−1(p) bna−1(p)

−ana(p) bna(p)

y

u

. (29)

Then by using (27a),y can be eliminated from (29) yielding the LPV-SS representation:

A(p) B(p)

C(p) D(p)

=

−a1(p) 1 0 . . . 0 b1(p)− a1(p)b0(p)... 0

. . . . . ....

......

.... . . . . .

......

−ana−1(p) 0 . . . 0 1 bna−1(p)− ana−1(p)b0(p)

−ana(p) 0 . . . . . . 0 bna(p)− ana(p)b0(p)

1 0 . . . . . . 0 d(p)

, (30)

whered = b0. This SS realization is minimal in the SISO case and has only static dependence.

In the MIMO case, by computing a set of maximally independentrows of [A,B] in (30), which

corresponds to independent state-variables, minimal realization via this approach also follows.

B. Augmented SS form

Assume that the LPV-IO model is given in the form

y +

na∑

i=1

ai(q−1p)q−iy =

nb∑

j=1

bj(q−1p)q−ju. (31)

Note thatai andbj have a special form of dynamic dependence which again can be introduced

into the parametrization of most of the available LPV-IO identification approaches. It is also

important that there is no feedthrough term, i.e.b0 = 0. Under these conditions, an augmented

equivalent LPV-SS representation is given in a straight forward manner as (32). Note that (32) is

not minimal, but has only coefficients with static dependence. This type of augmented SS form

is widely known in the LTI literature and also used in the identification toolbox in MATLAB

[20]. As a next step, an LPV model reduction algorithm, see Section V, can be applied to reduce

the state dimension, but preserve the static dependence. Based on Section IV-D, it holds true that

the minimal state dimension allowing static dependence of the SS form is larger than or equal

to the actual order of the system. Moreover, if the original IO model is already identified in the


15

form of (3), then this approach is still applicable as the resulting LPV-SS realization will only

be dependent onqp. If p is measurable by a faster sampling rate thanu andy, or the variation of

p is much slower than the sampling rate ofu andy, then dependence onqp does not introduce

complexity into the model in terms of control design.

A(p) B(p)

C(p) D(p)

=

−a1(p) . . . −ana(p) b2(p) . . . bnb(p) b1(p)

1 0 . . . . . . . . . 0 0...

. . . . . . . .. . . ....

...

0 . . . 1 0 . . . 0 0

0 . . . 0 0 . . . 0 1

0 . . . 0 1 . . . 0 0...

. . . . . . . .. . . ....

...

0 . . . . . . . . . . . . 1 0

1 0 . . . . . . . . . 0 0

. (32)

C. Observability form

Suppose that the LPV-IO model is given in the form

y +

na∑

i=1

ai(q−nap)q−iy = bna(q

−nap)q−nau. (33)

Note thatai and bj have a special form of dynamic dependence and the input has a delay of

q−na. Based on the realization theory of observability canonical forms (see [7]), an equivalent

SS realization reads as (34). Note that this LPV-SS realization is minimal in the SISO case and

has only static dependence. In the MIMO case, a minimal form of (34) can be computed via

finding the basis of its observability matrix following Young’s selection scheme II and computing

a state-transformation to obtain a MIMO observability canonical form, see [7, Ch.4].

A(p) B(p)

C(p) D(p)

=

0 1 0 . . . 0 0... 0

. . . . . ....

......

.... . . . . .

......

0 0 . . . 0 1 0

−ana(p) −ana−1(p) . . . . . . −a1(p) bna(p)

1 0 . . . . . . 0 0

. (34)


16

D. Elimination of dynamic dependence via state-transformation

In case the previous schemes can not be applied, the realization approach (13a-b) in Theorem

1 can be used, followed by a search for a state transformationwhich simplifies the coefficient

dependence onp to a static relation. In this way, it can be clearly characterized what is the

minimal achievable complexity of the LPV-SS realization ofthe given model. Such an approach

is tempting from the theoretical point of view but as we will see it has a significant computational

load.

In order to avoid detailed mathematical treatment of the problem, we consider the above given

idea in a simple case. For a given SISO LPV-IO representation(5) with na = 2 and0 ≤ nb ≤ na,

an LPV-SS realization of (5) is obtained in the form of

A B

C D

⋄ p =

α11 α12 β1

α21 α22 β2

γ1 γ2 δ

⋄ p (35)

where the coefficientsα11, . . . , δ ∈ R can have dynamic dependence onp. Note that in terms

of (11), no state transformation can annihilate dynamic dependence inD, which implies that a

SS equivalent of (35) exists only in the case ifδ has static dependence. The latter condition for

the realization of an LPV-IO model in the form of (3) is automatically satisfied. Consider

M1 =

m11 m12

m21 m22

, M2 =

µ11 µ12

µ21 µ22

, (36)

being full rank matrix functions inR2×2 such that

A(p) = (M1AM2) ⋄ p, B(p) = (M1B) ⋄ p, (37a)

C(p) = (CM2) ⋄ p, I = (←−M 1M2) ⋄ p, (37b)

whereA, B, C are matrix functions with staticp-dependence and←−· means backward time-shift

in the coefficient dependence like←−M 1 ⋄ p = M1 ⋄ (q−1p). The conditions (37a-b) imply the

existence of a state-transformation such that the resulting SS form with matrices{A, B, C, D}has only static dependence. Furthermore, (37a-b) corresponds to a system of bilinear equations.

These bilinear equations can be symbolically solved if it isassumed thatA, B, C are given

in a canonical form. Consider the example in Section II-C where we developed an LPV-SS


17

realization (15) via (13a-b) of the LPV-IO model (14). By solving (37a-b) w.r.t. (15) we can

arrive at the equivalent LPV-SS representation

A B

C D

⋄ p =

−p 1 0

p 0 1

p 0 0

with M1 =

0 1qp

1q2p

0

, M2 =

0 qp

p 0

, (38)

where the state-space matrices have only static dependence. This demonstrates that in some cases

it is possible to eliminate the dynamic dependence by using simple state-transformations.

In case no solution exists for (37a-b), it means that no minimal LPV-SS realization of the

model exists with static dependence. In this situation,

M1 =

m11 m12

m21 m22

m31 m32

, M2 =

µ11 µ12 µ13

µ21 µ22 µ23

, (39)

can be considered whereM1 is full column andM2 is full row rank, increasing the freedom of

the coefficient transformations by introducing an extra state variable. Then, solvability can be

checked again. Note that due to the increased number of state-variables, uniqueness of the

solution (if exists) in terms ofM1 and M2 is not guaranteed. In case of no solution, the

augmentation can be continued till feasibility or when the number of state variables exceeds

a limit.

Proposition 1: Consider (3) and assume that an LPV-SS realization of (3) is given in the

form of (7a-b) with dynamic dependence. If no state-transformation can be found that leads to

an equivalent SS form of (7a-b) with2na state-variables and with only static dependence, then

an exact SS realization of (3) with static dependence is not available.

The proof of the above given proposition is based on the fact that SS realization of (3) can only

introduce maximumna-order of time shifts in the dependence of the coefficients. If this order

of dynamic dependence can not be annihilated viana extra state variables, then it is also not

possible by usingna+1 state variables, hence an equivalent SS realization with static dependence

does not exist for (3).

It is an important remark that in general it is absolutely notguaranteed that an arbitrary

LPV-IO model has an SS realization with static dependence. Moreover, symbolic solution of the

state-transformation problem is a computationally demanding operation with an exponentially


18

increasing memory and computational load. Therefore, in terms of practical use, elimination of

dynamic dependence via state transformation is limited to small scale problems.

V. LPV MODEL ORDER REDUCTION

As we could see in the previous section, in some cases it is possible to obtain an equivalent

LPV-SS realization of a given LPV-IO model such that the resulting state matrices have only

static dependence. However, in some of these situations, the price we need to pay for such a

realization is the introduction of extra state variables. Often an approximation of such LPV-

SS models with a more economical size of state dimension is required in order to decrease the

complexity of control synthesis. So the natural question that rises is how we can efficiently apply

model reduction of the resulting realizations in order to eliminate non-significant dynamics if

needed.

A. Overview of LPV model reduction

Model reduction of LPV systems has been already studied in the literature since the middle

1990s and both exact and approximative reduction techniques have been developed. Exact

reduction techniques, such as [21], aim at a Kalman type of decomposition of the model to

a minimal and an eliminatable part preserving IO equivalence. However, in case of identified

LPV models, possibly with some over-parametrization, suchdecomposition is ill-posed due

to the effect of noise on the estimates. On the other hand, some of the discussed realization

approaches introduce extra-states to achieve static-dependence of the resulting LPV-SS form,

hence such states are not eliminatable without using dynamic dependence. This implies that

often only approximative methods can be used to obtain an economical model order in the

considered problem setting. In terms of approximative approaches, LTI reduction techniques,

such as coprime factor [22], optimal Hankel norm and balanced truncation [23] methods, are

extended to LPV systems and some of these approaches are implemented in the Enhanced

LFT toolbox [24], [25]. Model reduction based on balanced truncation is a well grounded

scheme in theory and most often used [26]. However, most common and practical LPV model

reduction techniques that are based on balanced truncationrequire quadratic stabilizability and

detectability of the full order model and they can not easilydeal with unstable models [23].

Both are important restrictions as quadratic stabilizability and detectability of an LPV model are

not always guaranteed and a major application area of systemidentification is to obtain models


19

of general unstable plants operated in closed loop. In the LTI case, the basic model reduction

methods in combination with coprime factorization [27] or spectral decomposition techniques

[28], can be used to reduce unstable systems. However extending such methods to LPV systems

is accompanied by a significant increase of the complexity ofthe reduction technique [22].

Thus, conventional model reduction approaches are often not applicable to reduce the state-

dimension of the SS realization of identified LPV models. Next, we investigate an alternative

model reduction method to overcome these restrictions.

B. Hankel matrix

In the sequel, we introduce a model reduction approach for MIMO LPV-SS representations

with affine dependence onp. The proposed approach is an extension of the Ho-Kalman realization

algorithm well known in the LTI case, see e.g. [29], and it is different from optimal Hankel norm

and balanced truncation approaches [23], [24]. The method is proposed as a tool to assist in

obtaining an economical-size LPV-SS realization of identified LPV-IO models. Unlike other LPV

model reduction techniques, the proposed technique here, in addition to its simplicity, does not

necessitate quadratic stabilizability or detectability of the full order model and it can be employed

for both stable and unstable systems without imposing any modification.

Consider the LPV-SS representation (1a-b), with affine dependence of the system matrices:

A(p) = A0 +

nψ∑

i=1

Aiψi(p), B(p) = B0 +

nψ∑

i=1

Biψi(p), (40a)

C(p) = C0 +

nψ∑

i=1

Ciψi(p), D(p) = 0, (40b)

whereψi(�) : P→ R are analytic functions onP and{Ai, Bi, Ci}nψi=0 are constant matrices with

appropriate dimensions. Furthermore, for well-possedness it is assumed that{ψi}nψi=1 are linearly

independent onP and normalized w.r.t. an appropriate norm or inner product.Define

M1 = [ B0 . . . Bnψ], Mj = [ A0Mj−1 . . . AnψMj−1 ]. (41)

Inspired by [30], [31], introduce the so calledk-step extended reachability matrix of (1a-b) as

Rk = [ M1 . . . Mk ], (42)

whereRk ∈ Rnx×(nu

∑kl=1(1+nψ)

l). The matrixRk is called the extended reachability matrix, as

similar to the LTI case, full rank ofRnXis a necessary condition for reachability of (1a-b),

however, opposite to the LTI case, it is not a sufficient condition [7], [32]. Let


20

Pk = [ 1 ψ1(p(k)) . . . ψnψ(p(k)) ]⊤, (43a)

Kk|i = Pk ⊗ . . .⊗ Pk−i ⊗ Inu , (43b)

Mki = diag(Kk−i|0, . . . , Kk−1|i−1), (43c)

where⊗ stands for the Kronecker product andIn is the identity matrix with dimensionn. Now

it is true that ifx(k − i) = 0 with k > i ≥ 0, then

x(k) = RiMkiUki, (44)

whereUki = [ u⊤(k − 1) . . . u⊤(k − i) ]⊤. From (44) it is obvious that for reachability, full-

rank ofRnXMknX

is required for allk and every possible trajectory ofp. Hence full-rank ofRnX

corresponds to reachability only in a structural sense. Now, define

N1 = [ C⊤0 . . . C⊤

nψ]⊤, Nj = [ (Nj−1A0)

⊤ . . . (Nj−1Anψ)⊤ ]⊤, (45a)

Lk|i = Iny ⊗ (Pk ⊗ · · · ⊗ Pk−i)⊤, (45b)

Nki = diag(Lk−i|0, . . . , Lk−1|i−1). (45c)

Based on similar considerations as before, introduce thek-step extended observability matrix as

Ok = [ N⊤1 . . . N⊤

k]⊤ (46)

whereOk ∈ R(ny

∑kl=1(1+nψ)

l)×nx. Now it is true fork > i ≥ 0 that

Yki = NkiOix(k − i), (47)

whereYki = [ y⊤(k − i) . . . y⊤(k − 1) ]⊤. Therefore,

Hij = OiRj ∈ R(ny∑il=1(1+nψ)

l)×(nu∑j

l=1(1+nψ)l), (48)

can be considered as the extended Hankel matrix of (1a-b). IfnP = 1, then

Hij =

C0B0 C0B1 C0A0B0 C0A0B1 . . .

C1B0 C1B1 C1A0B0 C1A0B1 . . .

C0A0B0 C0A0B1 C0A20B0 C0A

20B1 . . .

C1A0B0 C1A0B1 C1A20B0 C1A

20B1 . . .

C0A1B0 C0A1B1 C0A1A0B0 C0A1A0B1 . . ....

......

.... . .

For sufficiently largei andj, rank(Hij) = n, which can be considered as the McMillan degree

of (1a-b), in case of a minimal representationn = nx.


21

C. LPV Ho-Kalman algorithm

To construct a minimal state-space realization of (1a) or (32) with static dependence we can

proceed as follows. A Hankel matrix of the representation for sufficiently large dimensions is

constructed such thatrank(Hij) = n holds true. Then for any (full rank) matrix decomposition:

Hij = H1H2, (49)

with constant matricesH1 ∈ R(ny

∑il=1(1+nψ)

l)×n and H2 ∈ Rn×(nu

∑j

l=1(1+nψ)l) satisfying

rank(H1) = rank(H2) = n, there exist matrices functionsA(p), B(p), C(p) defined as in (40a-

b), such that thei-step observability matrixOi and thej-step reachability matrixRj generated

from their constant matrices satisfy

H1 = Oi, H2 = Rj . (50)

The matricesA(p), B(p), C(p) can be computed as follows: When givenH1 = Oi, the matrices

[ C⊤0 . . . C⊤

nψ]⊤ can be extracted by taking the firstny(1+nψ) rows and when givenH2 = Ri,

the matrices[ B0 . . . Bnψ] can be extracted by taking the firstnu(1 + nψ) columns. The

matrices{A0, . . . , Anψ} can be isolated by using a shifted Hankel matrix←−H ij, which is simply

obtained from the original Hankel matrixHij , by shifting the matrix one block column, i.e.

nu(1 + nψ) columns, to the left. It can be directly verified that←−H ij can be written as

←−H ij = Oi[ A0 . . . Anψ ](I1+nψ ⊗Rj−1), (51a)

= H1[ A0 . . . Anψ ](I1+nψ ⊗ H2), (51b)

whereH2 is generated fromH2 by leaving out the lastnu(1+nψ)j columns. Under the assumption

that j is large enough that rank(H2) = n, there exist pseudo inverse matricesH†1 and H†

2, such

thatH

†1H1 = I, H2H

†2 = I. (52)

These matrices can be computed asH†1 = (H⊤

1 H1)−1H⊤

1 and H†2 = H⊤

2 (H⊤2 H2)

−1. As a

consequence, it holds that

H†1

←−H ij(I1+nψ ⊗ H†2) = [ A0 . . . Anψ ]. (53)

A reliable procedure to compute the full rank decompositionof Hij is to use thesingular value

decomposition(SVD):

Hij = UkΣkV⊤k , (54)


22

whereUk andVk are unitary matrices andΣk is a diagonal matrix with positive entriesσ1 ≥· · · ≥ σk referred to as singular values. The choiceH1 = UkΣ

12k , H2 = Σ

12k V

⊤k directly leads

to H†1 = Σ

− 12

k U⊤k , H†

2 = VkΣ− 1

2k . By truncating the resultingH2 to generateH2 and H†

2, the

matrices{A0, . . . , Anψ} can be calculated according to (53).

As in the LTI case, it is demonstrated here that the LPV Ho-Kalman algorithm can be applied

to reduce the order or to find a minimal realization with a finite number of Markov parameters,

provided that certain rank conditions on a finite Hankel matrix are satisfied. Furthermore for

asymptotically stable systems, it can be shown that the state-space model obtained using the

LPV Ho-Kalman algorithm with SVD is balanced, i.e. the controllability and the observability

gramians of the state-space representation are equal, see [29] for more details. Theoretical error

bounds to characterize the approximation error of the reduced models has not been developed

yet, remaining to be the objectives of future research on this approach. However, numerical

approximation of the error between the reduced and the original plant is computable in aµ-test

sense with the Enhanced LFT toolbox [24], [25]. An additional property of the algorithm is that

with increasingi andj, the size ofHij exponentially increases while increasingnψ results in a

polynomial growth ofHij . This means that numerical problems and memory limits can quickly

play a limiting factor if the algorithm is applied for large scale models. Additional numerical

problems can also arise in case of unstable systems when too high powers of{A0, . . . , Anψ} can

result in a badly conditioned Hankel matrix. The curse of dimensionality, which is a drawback

of the algorithm, and the associated numerical problems canbe reduced by applying a kernel

based approach like in [31], numerical regularization, block-SVD algorithms, or Krylov subspace

methods [33].

Example 4:Consider an LPV-IO model in the form of (31) withna = 3, nb = 2, P = [−2π, 0]and

a1(q−1p) = (0.24 + 0.1q−1p), a2(q

−1p) = (0.6− 0.1√

−q−1p), (55a)

a3(q−1p) = 0.3 sin(q−1p), b1(q

−1p) = (1.25− q−1p), (55b)

b2(q−1p) = −(0.2 +

√

−q−1p), (55c)

By calculating the augmented SS form, a 4th order non-minimal LPV-SS representation results

as shown in (32). To compute a lower order representation of the produced LPV-SS model, the

LPV Ho-Kalman algorithm is applied.


23

According to (55a-b), the state matrices of the obtained LPV-SS model are written in terms

of (40a-d) withnψ = 3, ψ1(p) = p, ψ2(p) =√−p andψ3(p) = sin(p). Next the 3-step extended

reachabilityR3 as well as the 3-step extended observabilityO3 matrices are constructed in

terms of (42) and (46), respectively. These matrices are sufficient to perform model reduction

by the introduced algorithm. Continuing, the extended Hankel matrixH33, see (48), is computed

and an SVD is performed to decompose the Hankel matrix intoH1, H2 and to obtain their

pseudo inverses as in (49), (54). The singular values of the diagonal matrixΣ3, see (48), are

[2.6253, 1.3759, 1.0136, 0.382, 0, . . . , 0]. It is clear that the fourth singular value is less

significant than the rest, which suggests that it is possibleto approximate the model with a third

order one. Next, the shifted Hankel matrix←−H33 is determined and the matrixH2 is generated.

Finally the matrices{B0, . . . , Bnψ , A0, . . . , Anψ , C0, . . . , Cnψ} can be calculated using (53). Here

Ai ∈ R3×3, Bi ∈ R

3×1, Ci ∈ R1×3, i = 1, . . . nψ. 3D plots of theFrozen Frequency Response

(FFR) of the LPV-IO model (frequency response of the model for constantp(k), i.e.p(k) = p for

all k ∈ Z.) and the difference between it and the reduced model for allvalues ofP are shown in

Fig. 1. The approximate LPV-SS model preserves the static dependence on the scheduling signal

with the same scheduling functions, furthermore it gives balanced realization with acceptable

representation error (see Fig. 1). In addition, the reducedLPV-SS model is compared with the

LPV-IO model in terms of theBFR of simulated output over a set of 500 random trajectories of

u andp with arbitrary fast variation ofp. The minimum and the maximum values ofBFR were

84.21% and 94.75%, respectively. This justifies that the obtained reduced model via the LPV

Ho-Kalman algorithm gives an acceptable approximation of the original system.

VI. SIMULATION EXAMPLE (CHARGE CONTROL)

In this section, the methods discussed in Sections IV and V are tested and compared on

an application-based simulation study. The example considered here is the air charge control

problem of aspark ignition(SI) engine.

A. The engine manifold

The intake manifold of a SI engine for air charge control has ahighly nonlinear nature. It is

not an isolated system but part of the overall car model, Fig.2. The opening of the throttle valve

in the intake manifoldαlim is used to control the amount of the normalized air chargemnac. The

speed of the vehiclep2 influences the internal dynamics of the intake manifold and the engine


24

−6−4

−20

100

102

0

5

10

pFreq.(rad/sec)M

agni

tude

(dB

)

−6−4

−20

100

102

−150

−100

−50

0

pFreq.(rad/sec)

Mag

nitu

de (

dB)

Fig. 1. 3D plot of the frozen frequency response functions ofthe LPV-IO model for all values inP (top) and the difference

between these responses and the frozen frequency response functions of the reduced model (bottom) (Example 4).

p1engine vehicle

p2

v

+KP

−+Nref Te

Overall system

manifoldintake

αlim mnac

Fig. 2. Overall system including the engine and the vehicle in a closed-loop setting.

itself. The vehicle model, shown in Fig. 2, has an integral behavior, thus a negative feedback

is applied betweenp2 andαlim through a proportional gainKp = 3110

in order to stabilize the

engine speed.

A parameterized nonlinear physical model of the overall process was provided and experi-

mentally validated by the IAV GmbH company, Gifhorn, Germany; see [34] for more details.

Constructing an LPV model from the physical model has two drawbacks:

1) It necessitates approximation of some nonlinear characteristics in an ad-hoc manner which

reduces the accuracy of the derived LPV model.

2) It provides a continuous-time LPV model, and digital implementation of a controller

designed in continuous-time leads to performance deterioration due to hardware constraints

on the sampling time, hereTs = 0.01s.


25

Therefore LPV identification based on measured data can be used to identify a valid description

of the system for which the efficient machinery of LPV controlsynthesis can be applied.

B. LPV-IO Identification/Optimization

The intake manifold dynamics is affected by the pressurep1, which is generated inside the

manifold and the engine speedp2. Therefore, both signals are considered as scheduling signals

in the intended LPV model. Using data collected in the closed-loop setting of Fig. 2, different

LPV-IO models are identified for the intake manifold of the SIengine with:

1) static-dependence on the scheduling signals:MIO(p) as in (3).

2) dynamic-dependence on the scheduling signals:MIO(q ⋄ p) as in (26).

3) one-step backward dependence on the scheduling signals:MIO(q−1p) as in (31).

4) na-step backward dependence on the scheduling signals:MIO(q−nap) as in (33).

All the coefficient functions are chosen as second-order polynomials in p1 and p2 with zero

coefficients for the cross terms, e.g. in case of the modelMIO(p), which has static dependence,

the coefficient functions are given fori = 1 . . . , na, j = 1, . . . , nb as

ai(p) = ai0 + ai1p1 + ai2p21 + ai3p2 + ai4p

22, (56a)

bj(p) = bj0 + bj1p1 + bj2p21 + bj3p2 + bj4p

22. (56b)

It is known that the intake manifold can be well represented by a second-order model, therefore

na andnb are both chosen to be 2. In the system identification stage, the constant coefficients

{ail}2,4i=1,l=0 and{bjl}2,4j=1,l=0, see (56a-b), are estimated by a least-squares estimate. Itis known

that applying a direct estimate in closed loop via a least-squares approach results in a biased

model. However, there exist many methods to overcome such anissue by applying instrumental

variables, nonlinear optimization based techniques, etc.It should be noted that in the studied case

here we do not take into account any noise source as our aim is to test whether the different model

structures specified above are able to approximate the intake manifold dynamics. Therefore, the

bias issue does not play a role in this setting. Hence identification is used as an optimization

tool instead of estimation in a stochastic sense.

All LPV-IO model structures in (3), (26), (31) and (33) can bereformulated as

y(k) = φ⊤(k)θ, (57)

where, for instance, with static dependence we have


26

φ(k) = ϕ(k)⊗ P (k), ϕ(k) = [ −y(k − 1) −y(k − 2) u(k − 1) u(k − 2) ]⊤

P (k) = [ 1 p1(k) p21(k) p2(k) p22(k) ]⊤

and the constant coefficients are collected in the vectorθ ∈ R20.

In order to gather informative signals for optimizingθ, a white noiseNref , see Fig. 2, is

designed with uniform distributionU(760, 6250) and level change at random instances, which are

specified by an additional random variableU(0, 1) for deciding when to change the level. Another

signalv, see Fig. 2, with the same properties, but withU(10, 90), is designed to excite the input-

output dynamics of the system (fromαlim to mnac), see [35] for the probability characteristics of

this class of signals. As a consequence, the pressure signalis internally generated. The required

operating ranges for the different variables to design the input signal are as follows

y = mnac ∈ [10, 90]%, u = αlim ∈ [0, 100]%,

p1 ∈ [99, 950] hPa, p2 ∈ [760, 6250] rpm.

The input-output data is collected and divided into estimation and validation sets.

In case of a measurement noise, (57) corresponds to the one-step ahead prediction of the

output in a prediction-error identification setting under the assumption of an ARX noise model.

Here we consider the noiseless case in which (57) gives the data equation of the model directly.

Hence, for a set of input, scheduling signals and output data

ZN :={

[u(k), p1(k), p2(k), y(k)], k = 1, . . . , N}

;

the least-squares parameter estimate for (57)

θN := arg minθ∈R20

VN(θ, ZN), where VN (θ, Z

N) :=1

N

N∑

k=1

(y(k)− φ⊤(k)θ)2, (58)

can be obtained by linear regression.

The above given procedure has been adopted to calculate LPV-IO models for the intake

manifold with different structures. Table I shows a comparison between the resulting models

in terms of the BFR of the simulated outputs, where all of these models are simulated using

the same set of validation data. It is clear that the obtainedmodels with the structures (3),

(26), (31) and (33) give almost the sameBFR ≈ 95% which shows that all these structures are

approximating well the intake manifold plant.


27

TABLE I

BFR OF THE ESTIMATED MODELS FOR THE INTAKE MANIFOLD PLANT.

MIO(p) MIO(q ⋄ p) MIO(q−1p) MIO(q

−nap)

BFR % 96.47150 95.72519 96.03974 93.95923

C. LPV-SS Realization

Next, the identified LPV-IO models are transformed to LPV-SSforms by using the approaches

introduced in Section IV. Regarding the modelMIO(q−1p), a non-minimal 3rd-order LPV-SS

model is produced via the augmented SS form. Therefore, the Ho-Kalman algorithm is employed

to obtain a reduced 2rd-order LPV-SS model. It is worth to mention that the obtainednon-minimal

LPV-SS model is quadratically not stabilizable, which means that model order reduction based

on balanced truncation with fixed similarity transformation like [23] can not be used to reduce

the order of the model. In addition, without quadratic stabilizability and detectability properties,

simple and practical LPV controllers based on quadraticH∞ performance, e.g. [36], cannot

be synthesized. Interestingly, the resulting reduced LPV-SS model based on the Ho-Kalman

algorithm is balanced and quadratically stabilizable and detectable; this means that the lack of

quadratic stabilizability of the non-minimal model is due to the ’additional’ state, which has

been removed by the Ho-Kalman algorithm. However the reduced LPV-SS model approximates

the original model only withBFR = 75.58%.

Next, LPV controllers are synthesized based on all the transformed LPV-SS models, and the

resulting controllers are implemented on the original nonlinear model of the intake manifold. This

step is performed in order to assess the approximation quality of the different model structures

and the different input-output to state-space transformations methods in terms of control design.

D. Control Design

The linear matrix inequalities optimization-based LPV controller synthesis technique proposed

in [36] is used to design controllers for the transformed LPV-SS models with a mixed-sensitivity

loop shaping technique. This method is based on a polytopic representation of the LPV-SS

models, therefore an affine representation of the system matrices as in (40a-b) is necessary. The

performance requirements for the closed-loop are specifiedas:

• a rise time oftr = 0.15s, a settling time ofts = 0.3s.


28

TABLE II

BEST INDUCEDℓ2 GAIN FOR EACH DESIGN.

KMIO(p) KMIO(q⋄p) KMIO(q−1p) KMIO(q−nap)

γ 20.3987 1.1205 1.1091 2.8549

• overshootMp < 5%, steady state errore∞ < 1%.

• constraint on actuator usage.

These objectives are translated into shaping filters

WS =0.02z + 0.02

z − 0.9998, WKS =

0.000643z − 0.0002439

z + 0.9956,

to shape respectively the sensitivity and the control-sensitivity of the closed-loop. With these fil-

ters a generalized plant formulation of the models can be constructed. For consistent comparison,

the same shaping filters are used for all synthesis problems.Then the mixed-sensitivity criterion

is minimized such that the inducedℓ2 gain, of both the sensitivity and the control-sensitivity,is

less than some prescribed valueγ > 0. According to these considerations, controllers of order

five have been computed and the best achieved performance index γ for each design is given

in Table II. The controllerKMIO(p) is based on an LPV-SS model converted from the IO model

MIO(p) using the LTI rules. As a common property of the LMI-based synthesis approaches,

the order of any of the designed controllers is equal to the order of the generalized plant that

includes in addition to the system, the weighting filters anda first order pre-filter, which is used

here since theB matrix is parameter dependent, see [36] for more details.

E. Evaluation

Finally, all controllers are applied to the nonlinear physical model of the intake manifold.

Fig. 3a-e demonstrate the tracking of the normalized air charge rl = mnac at different levels

and periods of a specified typical trajectory with the different controllersKMIO(p), KMIO(q⋄p),

KMIO(q−1p) andKMIO(q−nap), respectively. In general, with all designed controllers,exceptKMIO(p),

mnac follows the given reference trajectory in a satisfactory manner, with a rise time, maximum

overshoot and steady state error within the limits which have been specified above. Even the

controllerKMIO(q−1p), which has been designed based on the worst approximate model, namely

the reduced augmented model, shows reasonable tracking with acceptable transient response,


29

see Fig. 3c. The best performance is achieved byKMIO(q⋄p). This is not surprising as the model

MIO(q ⋄ p) have a high quality fit of95% and the used SS realization is exact, while in the

MIO(q−1p) case the order reduction decreased the quality to75%. The second best result is

provided byKMIO(q−nap) due to the fact that the initial fit ofMIO(q−nap) was worse than

MIO(q ⋄ p). On the other hand, the high-quality modelMIO(p) based controllerKMIO(p), which

has been synthesized based on a SS model constructed by the LTI rules, violates the constraints

on tr, ts,Mp, e∞, see Fig. 3a. In order to achieve the performance of the othercontrollers by

KMIO(p), the shaping filters have been re-tuned and the best achievedperformance is shown in

Fig. 3e, which still violates the requirements onMp in addition to the undesired oscillations in

the control signal.

This example demonstrates the following: The proposed structures of LPV-IO models in (26),

(31) and (33) can provide a good approximation of the original system and can be used for

the purpose of control synthesis based on the associated realization approaches. Furthermore,

using the LTI rules to obtain an SS realization for an LPV-IO model may lead to an inadequate

approximation of the true dynamics, possibly resulting in asignificant performance loss of the

closed-loop control.

Based on the previous discussion the following practical advices can be given: In order to

ensure low complexity of the control design phase, LTI conversion rules to get the SS realization

should be tested first by adopting the criterion in Section III. If it turns out that the LTI rules

provide a poor SS realization, an LPV-IO model with any of thestructures in Subsection IV-A

or IV-C can be identified and via the proposed SS realization approaches an adequate LPV-SS

model of the plant can be obtained with static dependence. Asthese realizations are exact, initial

model fit indicates the expected performance of the designedcontroller on the true system. If

these structures do not result in an acceptable model for theunderlying process, then the structure

in Subsection IV-B can be adopted with the model reduction approach in Section V to provide

minimal SS realization on the possible expanse of model quality.

VII. CONCLUSIONS

An equivalent representation of an LPV-IO model in LPV-SS form, in general necessitates

dynamic dependence of the state-space realization on the scheduling signal. Neglecting this

fact can cause a significant performance loss in the control design phase. On the other hand,


30

18 20 22

12

14

16

18

r l [%]

48 50 52

20

30

40

50

58 60 62 64

20

30

40

time [s]68 70 72 74

10

20

30

40

(a) KMIO(p)

18 20 22

12

14

16

18

r l [%]

48 50 52

20

30

40

50

58 60 62 64

20

30

40

time [s]68 70 72 74

10

20

30

40

(b) KMIO(q⋄p)

18 20 22

12

14

16

18

r l [%]

48 50 52

20

30

40

50

58 60 62 64

20

30

40

time [s]68 70 72 74

10

20

30

40

(c) KMIO(q−1p)

18 20 22

12

14

16

18

r l [%]

48 50 52

20

30

40

50

58 60 62 64

20

30

40

time [s]68 70 72 74

10

20

30

40

(d) KMIO(q−nap)

18 20 22

12

14

16

18

r l [%]

48 50 52

20

30

40

50

58 60 62 64

20

30

40

time [s]68 70 72 74

10

20

30

40

(e) Improved controllerKMIO(p)

Fig. 3. Tracking ofmnac with the designed controllers.

the presence of dynamic dependence leads to difficulties in terms of controller design and

implementation. To deal with this problem, practical and systematic methods have been proposed.

First a criterion has been introduced to decide when the LTI conversion rules can be used without

serious consequences and how much can be lost in terms of model validity. Then a realization

algorithm was provided that is capable of yielding a minimalstate-space realization of any LPV-

IO model with the introduction of dynamic dependence. In order to avoid dynamic dependence

of the resulting SS realization, four pragmatic approacheshave been proposed to convert LPV-IO

models to LPV-SS realizations without introducing dynamicdependence. It was observed that


31

in some situations the restriction of static dependence in the realization results in a non-minimal

SS model. To obtain an economical SS representation of such augmented SS models even if

they are unstable or not quadratically stabilizable and detectable, an LPV Ho-Kalman type of

model reduction approach has been developed. Finally applicability and usefulness of all ideas

have been demonstrated by solving the charge control problem of an SI engine.

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